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Particle-in-Cell Simulations for Space Charge Effects

Explore the use of PIC simulations in ion traps and transport devices, with examples like gas catchers and high-intensity coolers. Learn about Poisson solver frameworks and particle motion integrators.

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Particle-in-Cell Simulations for Space Charge Effects

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  1. Particle-in-Cell Simulations for Studies of Space Charge Effects in Ion Trap and Transport Devices Ryan Ringle FRIB/NSCL Michigan State University R. Ringle, SMI 2019, 07/18/2019

  2. Outline • Motivation • Anatomy of a particle-in-cell (PIC) code • Example applications • Gas catcher (ACGS) • High-intensity cooler/buncher (ECB @ NSCL) • MR-TOF (Notre Dame) • Conclusions R. Ringle, SMI 2019, 07/18/2019

  3. Motivation – or why can’t I just use SimION? • SimION is an extremely versatile ion optics simulation suite • Supports complicated geometries • Easy user programming • Static magnetic fields • RF • Dielectrics • But… Basic charge repulsion effectsto estimate the onset of space-charge. New Poisson solver (Refine) support added in 8.1. (Note: for more advanced and numerically accurate space-charge and space-charge limited cathode emission calculations, under steady-state conditions, we suggest considering the CPO software.) R. Ringle, SMI 2019, 07/18/2019

  4. Motivation – or why can’t I just use SimION? • So what do I do if I want to simulate.. • Gas catchers with high incoming rates • Stopping fast ions generates a lot of He+/e- • Electrons have much higher mobility – removed faster than He+ • Arrive at a steady state to see the impact of space charge on applied electric fields • Extraction nozzles with gas flow and nA’s of ion current • Typically part of gas catcher extraction • Large amounts of stable ion current • RF carpets or funnels • RFQ cooler/bunchers for high intensity beams • Large numbers of ions increase phase space • Could lead to losses on extraction • Could lead to variable axial and transverse emittance • MR-TOF performance as a function of ion number R. Ringle, SMI 2019, 07/18/2019

  5. Anatomy of a PIC code • Main Components1 • Particle storage class • Stores info about particles in simulation • Poisson eq. solving routine • Many approaches (FEA, multigrid, SOA, FFT) • FFT are fastest but require special treatment of internal boundary conditions • Others are slower, but more flexible, support irregular geometries • Particle velocity and position integrators • May need to pick appropriate routine for damped motion, magnetic fields, etc. • Special routines • Capacity matrix for internal boundary points (FFT) • Buffer gas methods (hard sphere, SDS) initialize particle storage initialize optional routines initialize boundary conditions distribute charge on grid calculate electric potential and field Repeat in Steps of Dt update particle positions and velocities update boundary conditions 1R. J. Ringle, Int. J. Mass Spectrom. 303, 42 (2011) R. Ringle, SMI 2019, 07/18/2019

  6. Poisson solver example (3D cylindrical) • Poisson’s Eq. in cylindrical coordinates • r : charge source term (charged particles) • Assume Dirichlet (specified voltage) boundary conditions Since the potential is periodic in q it can be written as a truncated Fourier series : q Now insert into back into Poisson’s Eq. R. Ringle, SMI 2019, 07/18/2019

  7. Poisson solver example (3D cylindrical) • notice that the θ dependence has been removed • left with Nθ 2D equations to solve. z r1 • Choose a grid in the r-z plane (Nr, Nz) • Apply centered-difference approx. Nθ systems of linear equations that can be solved using standard methods1 qi y x 1B. L. Buzbee, G. H. Golub, and C. W. Nielson, SIAM Journal on Numerical Analysis (1970) R. Ringle, SMI 2019, 07/18/2019

  8. Distribution of Charge on a Grid • Many particles are tracked and are the source of charge term, ρ. • Charge must be assigned to the grid at every time step in order to update the total potential • Iterate over every simulation particle and assign charge to a grid point • Nearest grid point (easiest, less accurate) • Locate nearest grid point, assign all charge to that point • Crude assignment can cause discontinuities1 • Cloud-in-cell2 (computationally more expensive, more accurate) • Assign fraction of charge to nearest grid points based on volume weighting 1R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles (1988) 2C. K. Birdsall and D. Fuss, J. Comput. Phys. (1969). R. Ringle, SMI 2019, 07/18/2019

  9. Particle Motion Integrator Example Apply centered-difference approx. wrt time Apply Boris1 transformation and substitute Correction term for small error in cyc. frequency introduced by discrete time step 1J. P. Boris, Proceedings of the Fourth Conference on Numerical Simulation of Plasma 3 (1970) R. Ringle, SMI 2019, 07/18/2019

  10. Particle Motion Integrator Example Velocity advance in radial plane for B = Bz • Solve v+ in the radial plane • α = λωc*dt/2 Velocity advance in axial dimension • very simple (no B dependence) Position advance • Use calculated velocity components to update the particle position R. Ringle, SMI 2019, 07/18/2019

  11. Application Example : ACGS • Goal: Deliver high-intensity beams with highest efficiency • Main Body • Each stopped rare isotope generates ~ 106 He+/e- pairs • Slow He+ ion removal increases space charge in ACGS body, reducing transport efficiency • Extraction Orifice • Charge exchange and/or direct ionization of impurities in He buffer gas generates beams of stable molecules • Extraction efficiency can be compromised by large stable beam currents R. Ringle, SMI 2019, 07/18/2019

  12. Application Example : ACGS • 3D Cartesian geometry • Energy deposition from LISE++ • He+ and e- included • 108pps incoming 77Br • P = 200 Torr @ 300 K • Push = 40 V/cm V/cm Equilibrium reached at t ~ 2.2 ms R. Ringle, SMI 2019, 07/18/2019

  13. Application Example : ACGS • 2D cylindrical RZ geometry • Gas flow calculated with COMSOL • T = 60 K • Traveling wave transport • freq = 6.6 MHz • amp = 100 V • 10 nA of O2+ impinging on RF carpet • Orifice potential = 6 V orifice RF carpet cone R. Ringle, SMI 2019, 07/18/2019

  14. Application Example : ACGS • 10 nA of O2+ impinging on RF carpet • T = 60 K • Cone geometry offers better throughput R. Ringle, SMI 2019, 07/18/2019

  15. Application Example : High-Intensity RFQ Ion Beam Cooler Bunchers • Oscillating quadrupole electric fields provide transverse confinement – Radio Frequency Quadrupole (RFQ) • Confinement depends on RF voltage, frequency • Analogous to focusing/defocusing system • Cooling • Buffer gas (usually He) damps ion motion • Moving and trapping ions • Axial electric field created by segmentation of RFQ rods and application of additional DC potentials along rod in addition to RF • Apply longitudinal push/drag field or form axial trap RF 0° RF 180° Beam Path R. Ringle, SMI 2019, 07/18/2019

  16. Application Example : High-Intensity RFQ Ion Beam Cooler Bunchers • Crosscut geometry • Fewer electrodes • Simplified wiring • Smoother gradient • RF • Produced via tunable LC circuit • Impedance matched with transmission line transformer • Capacitively coupled to split electrodes • DC • Leads fed through inductor • Minimizes RF at power supply R. Ringle, SMI 2019, 07/18/2019

  17. Application Example : High-Intensity RFQ Ion Beam Cooler Bunchers • 3D cylindrical geometry • Buffer gas interactions via hard sphere scattering • Boundary conditions of extraction system obtained using SimION R. Ringle, SMI 2019, 07/18/2019

  18. Application Example : High-Intensity RFQ Ion Beam Cooler Bunchers • Cooling ions to steady state in RFQ • Ions injected over time • Distribution recorded at end of sim • Used in next stage of ejection sims • A = 39 • RF amp = 315 V • RF freq = 4 MHz • Pressure = 10-3 mbar • q = 0.4 • Radial trap depth = 32 V No space charge 108 ions R. Ringle, SMI 2019, 07/18/2019

  19. Application Example : High-Intensity RFQ Ion Beam Cooler Bunchers • Ions are ejected from RFQ and recorded at the end of the model space • Transverse emittance, time of flight, and energy distributions 108 ions x and y emittance measured using a pepperpot No space charge R. Ringle, SMI 2019, 07/18/2019

  20. Application Example : High-Intensity RFQ Ion Beam Cooler Bunchers Time-of-flight distributions Energy distributions No space charge 108 ions R. Ringle, SMI 2019, 07/18/2019

  21. Application Example : Notre Dame MR-TOF PIC Model Space - boundary conditions extracted from SIMION - 2D cylindrical geometry No space charge • Ion distribution : 50% 40 amu, 50% 40.02 amu • Record ion parameters as they pass through the center of the MR-TOF • Determine peak separation as a function of time • Peak coalescence rapidly sets in with around 25,000 ions R. Ringle, SMI 2019, 07/18/2019

  22. Conclusions • Space charge often has a negative impact on the performance of ion trap and ion transport devices • The PIC technique is well suited to exploring the subtleties of space charge and can help mitigate its impact • Gas catchers • Cooler/bunchers • MR-TOFs • For the most part you still have to build your own, particularly if performance is desired R. Ringle, SMI 2019, 07/18/2019

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